Brief review on Z γ -circle patterns corresponding to regular SG-circle patterns 69

and only if the corresponding circles intersect orthogonally. On these edges we define two conductance functionsC and ˜C by

C(e) =C(R(zj), R(zk)) :=

µR(z_j)

R(zk)+R(z_k) R(zj)

¶₋₁ , C(e) = ˜˜ C(r(zj), r(zk)) :=

µr(zj)

r(zk)+r(zk) r(zj)

¶₋₁ ,

where the edge e= [zj, zk] connects the verticeszj, zk∈V. Now estimations (4.16) imply that both positive functions C >0 and ˜C > 0 are uniformly bounded away from 0 (and from infinity). These two conductance networks on V⊂Z² can be continued to all of Z² by reflection in the lines{|M|=|N|}. From Theorem 4.9 we deduce that the two networks (Z², C) and (Z²,C) are reccurent.˜

Consider the following positive functions onV

f1(z) =r(z)/R(z)>0 and f2(z) =R(z)/r(z) = 1/f1(z)>0.

By Proposition 4.11 these functions are subharmonic, in particular X4

j=1

p(z₀, z_j)f₁(z_j) = X4

j=1

C(R(z0), R(zj)) P₄

j=1C(R(z0), R(zj))f₁(z_j)≥f₁(z₀) and X4

j=1

˜

p(z0, zj)f2(zj) = X4

j=1

C(r(z˜ 0), r(zj)) P₄

j=1C(r(z˜ 0), r(zj))f2(zj)≥f2(z0),

wherez1, z2, z3, z4are incident to the interior vertexz0∈V. Using the boundary conditions of the circle patterns, the above inequalities are also true iff1 andf2 are continued to all of Z² using reflection. Note that M−f1 andM −f2 are superharmonic for all constants M ∈R. Iff1 orf2is bounded from above, we thus get a positive superharmonic function using the upper bound. Then Theorem 4.10 implies that both functions are constant. Thus r≡Rand consequently both circle patterns coincide.

To conclude, we prove thatf1 andf2 are bounded from above. Denote byM1(n) and M2(n) the maximum off1andf2, respectively, for the set of vertices of thenth generation.

As f1 and f2 are subharmonic, they assume their maxima on the boundary. Therefore the functions M1 andM2 are monotonically increasing. The estimations (4.14) and (4.15) imply that the quotients of any two radii of one circle pattern in the nth generation are bounded from above for n≥2, as two vertices in thenth generation can be connected by at most 4n edges using only vertices of the nth andn+ 1st generation. So their quotient is bounded by e^4A for both radius functions r and R. Note that with the normalization r(0) = 1 =R(0), the maximaM1andM2are bounded from below by 1. Thus their product

M1(n)M2(n) =f1(z_M⁽ⁿ⁾₁)f2(z⁽ⁿ⁾_M2) =R(z⁽ⁿ⁾_M

1) r(z⁽ⁿ⁾_M1)

r(z_M⁽ⁿ⁾

2) R(z_M⁽ⁿ⁾₂)≤e^8A

is bounded from above. Herez⁽ⁿ⁾_M1 andz_M⁽ⁿ⁾₂ denote the vertices of thenth generation where f1andf2assume their maxima, respectively. ThereforeM1andM2are also bounded. This finishes the proof of uniqueness.

4.4 Brief review onZ^γ-circle patterns corresponding to

and edges connecting vertices z1, z2 ∈ V(SG) with |z1 −z2| = √

2. Let ψ ∈ (0, π) be a fixed angle. As in Section 3.6, denote by α_ψ the following regular labellingα_ψ on the edges E(SG). Let [z1, z2] ∈ E(SG) be an edge connecting the vertices z1, z2 ∈ V(SG).

Without loss of generality, we may assume that Re(z1)≤Re(z2). Then αψ([z1, z2]) =ψ if Im(z1)≤Im(z2) andαψ([z1, z2]) =π−ψ if Im(z1) ≥Im(z2). Using this notation, we consider the following generalization of Definition 4.3.

Definition 4.12 ([3, Definition 2]). For 0< γ <2, the discrete map Z^γ :Z²₊ →Cis the solution of

q(fn,m, fn+1,m, fn+1,m+1, fn,m+1) :=(fn,m−fn+1,m)(fn+1,m+1−fn,m+1)

(fn+1,m−fn+1,m+1)(fn,m+1−fn,m) = e^2i(ψ−π) (4.17) and (4.2) with the initial conditions

Z^γ(0,0) = 0, Z^γ(1,0) = 1, Z^γ(0,1) = e^γ(π−ψ)i.

Similar arguments as in the orthogonal case imply that one can again associate a circle pattern (for a quadrant ofSGcorresponding toZ²₊ and related toVand αψ) to the map Z^γ. Equation (4.3) is still satisfied and the corresponding version of equation (4.4) reads (see [3, Proposition 3]):

(N+M)(R(z)²−R(z+ 1)R(z−i)−cosψR(z)(R(z−i)−R(z+ 1)))(R(z+i) +R(z+ 1)) +(M−N)(R(z)²−R(z+i)R(z+1)−cosψR(z)(R(z+i)−R(z+1)))(R(z+1)+R(z−i)) = 0.

(4.18) The corresponding versions of equations (4.5) and (4.6) are similar. Furthermore, the following results generalize the orthogonal case.

Theorem 4.13 ([3]). (i) If R(z) denotes the radius function corresponding to the dis-crete conformal mapZ^γ for some0< γ <2, then it holds that

(γ−1)(R(z)²−R(z−i)R(z+ 1)−cosψR(z)(R(z−i)−R(z+ 1))≥0 (4.19) for allz∈V\ {±N+iN|N ∈N}.

(ii) For0< γ <2, the discrete conformal mapsZ^γ given by Definition 4.3 are embedded.

(iii) The circle patterns corresponding to the discrete mapsZ^γ for0< γ <2are embedded.

(iv) If R(z)denotes the radius function which corresponds to the discrete conformal map Z^γ for some 0< γ < 2, then R(z) = 1/R(z)˜ is the radius function corresponding to the discrete conformal mapZ^γ˜ for˜γ= 2−γ.

4.5 Uniqueness of quasicrystallic Z^γ-circle patterns

As explained in Section 13 of [14], discrete analogs of the power function z^γ can also be defined for quasicrystallic rhombic embeddings instead of Z²₊ (or Z²). In particular, let A={±a1, . . . ,±ad} ⊂S¹ be the set of edge directions. Suppose thatd >1 and that any two non-opposite elements ofAare linearly independent overR. For 0< γ <2 define the following values of the comparison functionwon the coordinate semi-axis ofZ^d₊:

w(nek) =







1 ifn= 0,

a^γ−1_k = e^{(γ−1) logak} ifnis odd, Q_n/2

m=1 m−1+^γ₂

m−^γ₂ ifn≥2 andnis even.

(4.20)

The value of the logarithm logak is chosen similarly as in the definition of the discrete Green’s function in Section 3.3. Using the Hirota Equation (3.12), this functionw can be extended to the whole sectorZ^d₊. Using suitable branches of the logarithm,wmay also be extended to other sectors or to a branched covering ofZ^d.

Note that for d = 2 the above definition leads to the same circle patterns as Defini-tion 4.12. Therefore, by Theorem 4.13 (ii), the circle pattens corresponding to the re-striction of w to combinatorial surfaces Z²₊ ⊂Z^d₊ which are spanned by two coordinate semi-axis are embedded. We can apply finite and infinite flips to obtain other monotone combinatorial surfaces corresponding to rhombic embeddings. In particular, we obtain re-strictions to Z^d₊ of the plane based combinatorial surfaces constructed in Examples 3.3 and 3.4. Lemmas 3.21 and 3.25 imply that these again lead to embedded circle patterns.

Thus we have

Theorem 4.14 (Embeddedness of quasicrystallic Z^γ-circle patterns). Let Ω ⊂ Z^d₊ be a simply connected monotone combinatorial surface. Then the circle pattern given by the function w with initial values (4.20)is embedded.

The aim of this section is to prove uniqueness of quasicrystallicZ^γ-circle patterns using the same arguments as for the orthogonal case. The following proposition is a simple generalization of Proposition 4.11 and requires a (geometric) restriction which ensures that the kites corresponding to intersecting circles are convex. Note that flips (finite or infinite) may destroy convexity, that is if all kites of a circle pattern are convex before performing a flip this is not necessarily true afterwards.

Proposition 4.15. LetD be a quasicrystallic rhombic embedding with associated graphG.

Let αbe the labelling corresponding toD. Let v0 be an interior vertex ofG with incident vertices v1, . . . , vm. Consider two circle patterns for Gand αwith radius functionsr and ρ. Denote rj =r(vj) andρj =ρ(vj) for j = 0,1, . . . , m and suppose that rj ≥r0cosαj

andρj ≥ρ0cosαj forj= 1, . . . , m, whereαj =α([v0, vj]). Then X4

j=1

cjrj

ρj ≥ r0

ρ0

X4

j=1

cj and

X4

j=1

cjρj

rj ≥ ρ0

r0

X4

j=1

cj, (4.21)

where cj= sinαj/((ρj/ρ0) + (ρ0/ρj)−2 cosαj)forj= 1, . . . , m.

Proof. The proof is similar to the proof of Proposition 4.11 and based on a Taylor expansion offα(x+ logy) abouty= 1.

fα(x+ logy) =fα(x) +f_α⁰(x)(y−1)− sinα(e^x+logξ−cosα)

2ξ²(cosh(x+ logξ)−cosα)²(y−1)² withξ=t+ (1−t)y for somet∈(0,1). Equation (2.2) for the two circle patterns implies

π= Xm

j=1

fαj(^r_r^j

0) =P_m

j=1fαj(log_ρ^ρj

0 + log^r_r^jρ0

0ρj)

= Xm

j=1

fαj(log^ρ_ρ^j

0)

| {z }

=π

+ Xm

j=1

f_α⁰_j(log^ρ_ρ^j

0) µrjρ0

r0ρj

−1

¶

− Xm

j=1

(^ρ_ρ^j

0ξj−cosαj) sinαj

2ξ²_j(cosh(ρj/ρ0+ logξj)−cosαj)² µrjρ0

r0ρj −1

¶₂ , where ξj =tj+ (1−tj)^r_r^jρ0

0ρj >0 with suitabletj∈(0,1) forj= 1, . . . , m. Furthermore ρj

ρ0

ξj−cosαj =tjρj

ρ0

+ (1−tj)rj

r0

−cosαj ≥0.

by our assumption. Thus Xm

j=1

f_α⁰_j(log^ρ_ρ^j

0) µrjρ0

r0ρj

−1

¶

≥0.

This implies the first claim sincef_α⁰_j(log(ρj/ρ0)) = sinαj/((ρj/ρ0) + (ρ0/ρj)−2 cosαj).

The second claim follows from the fact, that 1/ρ and 1/r are also radius function of circle patterns forGandαby Lemma 2.8. Also, the coefficientscj are invariant under the transformationρ7→1/ρ.

In order to apply this proposition, we need the following result on the convexity of the kites of theZ^γ-circle patterns. For the cases excluded below, there exist non-convex kites, because already the kite built from the circle centered at the origin is non-convex.

Lemma 4.16. If ψ ≥ π/2 and 0 < γ < 2, all kites in the Z^γ-circle pattern given by definition 4.12 are convex.

If ψ < π/2 and (π−2ψ)/(π−ψ)≤γ ≤π/(π−ψ), all kites in the Z^γ-circle pattern given by definition 4.12 are convex.

Proof. The claims on convexity are simple consequences of equation (4.18) and the results on embeddedness of Theorem 4.13 (i) and (ii).

In particular, forπ−ψ≤π/2 the kites with intersection angleψ≥π/2 at black vertices are always convex. From equation (4.18) and inequality (4.19) we can deduce by simple calculations that R(z+ 1) cos(π−ψ)≤R(z) andR(z) cos(π−ψ)≤R(z+ 1). Thus the remaining kites are also convex.

Ifψ < π/2 and (π−2ψ)/(π−ψ)≤γ ≤π/(π−ψ), the kites with intersection angle π−ψ > π/2 at black vertices are convex. In this case inequality (4.19) only implies that R(z+i) cosψ≤R(z) if 0< γ <1 andR(z) cosψ≤R(z+i) if 1< γ <2. This shows that for all kites with white verticesz andz+iand intersection angleψthe angle at the point corresponding toz for 0< γ <1 and toz+ifor 1< γ <2 respectively is smaller thanπ.

This excludes some types of non-convex kites, but not all.

For further use, note that the curves Γn of the Z^γ-circle pattern corresponding to regular SG-circle patterns which are constructed in an analogous way as in Section 4.2 have analogous properties due to Theorem 4.13 (i) and (ii). They are embedded without self-intersections and the vectorvn(m) rotates clockwise for 0< γ <1 and counterclockwise for 1< γ <2 along these curves.

Without loss of generality, we only consider the case 1< γ <2 further. For 0< γ <1 the proof is very similar. First, consider a kite on the symmetry axis, that is with white vertices corresponding to iK and i(K+ 1). Then the assumption γ(π−ψ) ≤π and the properties of the curves Γn imply that the angle of this kite at the vertex corresponding to i(K+ 1) is larger thanπ−2ψ. Consequently, the angle at the vertex corresponding to iK is smaller than 2π−(π−2ψ)−2ψ=π. Thus the kites on the symmetry axis are convex.

Next, we consider the intersection angles with the half linesR⁺ and e^{iγ(π−ψ)/2}R⁺ and of the lines in direction of the vectorvn(m) and of the vector ˆvm(n) of the mirror reflected curve ˆΓm. We additionally assume thatnis odd. See Figure 4.4 for an illustration and for the notation of the angles. As the kites on the symmetry axis are convex, we deduceα≤ψ andβ ≤π/2. Furthermore using the assumptionγ(π−ψ)≤πwe obtain

π

2 ≥β =π−γπ−ψ

2 −α≥π−π

2 −ψ=π 2 −ψ, π

2 =π−ψ−π

2 +ψ≥η=π−ψ−β≥π−ψ−π 2 =π

2 −ψ.

Finally, consider a kite with white vertexfn,m for oddnand m < n. We estimate the angles α1 and α2 at this vertex of the kites containing the points fn,m−1, fn,m, fn+1,m

R⁺ α

vˆm(n) vn(m) ψ e^{iγ(π−ψ)/2}R⁺ Γn β

η Γˆm

e^iγ(π−ψ)R⁺

γ^π−ψ₂ ψ

Figure 4.4: Geometric considerations for the vectors along the curves Γn for oddn.

R⁺ e^{iγ(π−ψ)/2}R⁺

ψ R⁺

e^{iγ(π−ψ)/2}R⁺

Γn

β4

β3

η2

η1

α1

β4

β6

η1

η2

β2

fn−1,m

γ^π−ψ₂ β1 γ^π−ψ₂

fn,m fn,m

fn,m+1

fn−1,m

fn,m−1

fn+1,m

fn,m−1

fn+1,m

fn,m+1

β5

β1

α2

Figure 4.5: Geometric considerations for the angles of the kites at fn,m for odd n and m < n.

andfn+1,m,fn,m,fn,m+1 respectively. Note that these kites both have intersection angles ψ. See Figure 4.5 for an illustration and for the notation. Using the above estimations we obtain β1≤ψ,β4≥π/2,β5≤π/2, andβ6≥π/2 and thus

α1= 2π−β1−β4−γπ−ψ

2 ≥2π−ψ−π

2 −ψ−π

2 =π−2ψ, η1=π−2π+β5+γπ−ψ

2 +β1≤ −π

2 +γπ−ψ 2 +β1, η2=π−β6−(π−β4)≤β4−π

2, α2=α1+η1+η2≤2π−β1−β4−γπ−ψ

2 −π

2 +γπ−ψ

2 +β1+β4−π 2 =π.

Therefore the angle at fn−1,m−1 of the kite containing the points fn,m−1,fn,m, fn+1,m is 2π−2ψ−α1≤π. So this kite is convex. Furthermore, we deduce that the kite containing the pointsfn+1,m,fn,m,fn,m+1is also convex. Consequently all kites containing the white vertex fn,mare convex. This shows that theZ^γ-circle pattern withγandψ satisfying the above assumptions only contains convex kites.

The rigidity of regular Z^γ-circle patterns can now be proven by similar arguments as for the orthogonal case.

Theorem 4.17 (Rigidity of regular Z^γ-circle patterns). If ψ ≥ π/2 and 0 < γ < 2 or ψ < π/2 and (π−2ψ)/(π−ψ) ≤ γ ≤ π/(π−ψ) then the Z^γ-circle pattern given by Definition 4.12 is the unique embedded regular SG-circle pattern for Z²₊ and αψ (up to global scaling) with the following properties.

(i) The infinite sector{z=ρe^iβ ∈C:ρ≥0, β∈[0, γ(π−ψ)]} with angle γ(π−ψ) is covered by the union of the corresponding kites of the circle pattern.

(ii) The centers of the boundary circles lie on the boundary half lines.

(iii) All kites corresponding to intersecting circles are convex.

The proof of Theorems 4.8 and 4.17 actually also shows the following generalization.

Theorem 4.18. Let ψ∈(0, π)and γ∈(0,2)∩[^π−2ψ_π−ψ,_π−ψ^π ]. DefineZ^γ-circle pattern on all four sectors Z±×Z± according to Definition 4.12 and glue these patterns such to a circle pattern Cγ on a cone with cone angle2πγ. Then any embedded circle pattern with the same combinatorics and intersection angles which covers the same cone with one center of circle placed at the apex and which has only convex kites coincides withCγ (up to scaling and rotation about the apex of the cone).

The following theorem is a direct consequence of this generalization.

Theorem 4.19(Rigidity of quasicrystallicZ^γ-circle patterns I). LetD be a quasicrystallic rhombic embedding of a b-quad-graph covering the whole plane. LetA={±a1, . . . ,±ad} ⊂ S¹ be the edge directions, whered >1and any two non-opposite elements of Aare linearly independent overR. Denote byψminthe minimum of the undirected angles between any two elements ofA. Letγ∈(0,2)with(π−2ψmin)/(π−ψmin)≤γ≤π/(π−ψmin). Assume that the origin is a white vertex ofD. Then a quasicrystallicZ^γ-circle patternCγ corresponding to D and embedded on a cone with cone angle 2πγ can be defined using the definition of the comparison functionwon the 2dsectors ofZ^d which contain the combinatorial surface Ω_D; see (4.20) and the remarks below. Assume further that the brickΠ(Ω_D)contains the whole lattice Z^d.

LetC be an embedded circle pattern with the same combinatorics and the same intersec-tion angles which covers the same cone with one center of circle placed at the apex. Extend the comparison function w for C from ΩD to Z^d. For each Z²-sublattice which contains two coordinate axes suppose that the corresponding circle pattern built according to this comparison function has only convex kites. Then C coincides with Cγ up to scaling and rotation about the apex of the cone.

Note that the assumption on the convexity of the kites is only a restriction for a (small) neighborhood of the origin. This is due to Lemma 3.33 (or Corollary 3.44) which implies that the ratio of the radii is almost one and thus the corresponding angles are almost as the same in the isoradial case if the combinatorial distance to the origin is big enough.

If all intersection angles of the labellingαtaken from the rhombic embedding are larger than π/2, then the kites of any corresponding circle pattern are convex. Note that this restriction is the same as for the hyperbolic maximum principle in Lemma 2.13. Examples of such rhombic embeddings are suitable regular hexagonal patterns as shown in Figure 2.3(b).

Hexagonal circle patterns and in particular analogs of the holomorphic mappingsz^γ have been studied by Bobenko and Hoffmann in [12]. In the case that all intersection angles are larger thanπ/2. Thus the proof of Theorem 4.8 can be directly adapted.

Theorem 4.20(Rigidity of quasicrystallicZ^γ-circle patterns II). LetDbe a quasicrystallic rhombic embedding of a b-quad-graph. Assume that the corresponding labellingα:F(D)→ [π/2, π)only has values larger than π/2. Assume further that the origin is a white vertex.

Let γ∈(0,2).

v1 v2 v3 v4

Figure 4.6: An illustration of the network obtained fromGby shortening.

Define a quasicrystallicZ^γ-circle patternCγ forD andαwhich is embedded on a cone with cone angle 2πγ using the definition of the comparison function on the 2d sectors of Z^dwhose union contains the combinatorial surfaceΩD; see(4.20). In particular, the circle corresponding to the origin is centered at the apex of the cone.

Let C be an embedded circle pattern forD andαwhich covers the same cone. Suppose that the circle corresponding to the origin is centered at the apex and thatC has only convex kites. Then C coincides withCγ (up to scaling and rotation about the apex of the cone).

In order to apply the same arguments as for the proof of Theorem 4.8, note that the simple random walk on the associated graph for an infinite rhombic embedding is recurrent.

Lemma 4.21. LetDbe a quasicrystallic rhombic embedding of a b-quad-graph which covers the whole plane C. LetG be the associated infinite graph built from white vertices. Then the simple random walk on Gis recurrent.

Proof. Without loss of generality we assume that all edges ofD have length one. SinceD is a quasicrystallic rhombic embedding, there is a constantC1>0 such that the area of any rhombus ofDis bigger thanC1(and smaller than one). Furthermore, the number of rhombi incident to a vertex ofDis uniformly bounded from above by a constantC2=C2(d), where dis the dimension ofD.

Without loss of generality we assume that the origin is a vertex ofG. In the following, we construct another graph G⁰ from Gby shortening. Recall that for the simple random walk all edges have conductance c(e) = 1 and also resistance r(e) = 1/c(e) = 1. If two incident vertices are identified, that is the conductance of the connecting edge is increased to ∞ and the resistance decreased to 0, then the effective resistance of the new network is certainly smaller. More generally, the proceedure of identifying a set of vertices of G, which corresponds to increasing the conductances of the edges between these vertices to

∞ (and to decreasing the corresponding resistance to 0), is calledshorteningand reduces the effective resistance; see [32, Section 2.2.2] or [77, Theorem (2.19)]. Remember that our aim is to prove that the effective resistanceReff of the networkGwith unit conductances is infinite. Therefore, it is sufficient to show that we have infinite effective resistanceR⁰_eff=∞ for a network G⁰ which is obtained fromGby shortening.

In particular, set %_k = 4k fork ∈N. Denote by V_k the set of vertices of G which are contained in the annulusAk={z∈C:%k−1≤ |z|< %k}fork≥1. ThenV(G) =∪^∞_k=1Vk. Identify the vertices of each Vk to one new vertex vk. Then by construction, vk is only incident tovk−1fork≥2 and tovk+1fork≥1. An illustration of the shortened network is shown in Figure 4.6. Denote by #Ek the number of edges which are incident tovkandvk+1

for k≥1. Then the effective resistance of the shortened network is R⁰_eff =P_∞

k=11/#Ek. Fork≥2 we have

#Ek ≤number of rhombi ofD which intersect the circle{{z∈C:|z|=%k}

≤F({z∈C:%k−2≤ |z|< %k+ 2})/C1

= (π(%k+ 2)²−π(%k−2)²)/C1=k^32π_C

1,

where F(·) denotes the Euclidean area. This implies that R⁰_eff = ∞. Thus the simple random walk on Gis recurrent.

Convergence for isoradial circle patterns

In this chapter we state and prove convergence theorems for isoradial circle patterns. First we proveC¹-convergence for a general class of isoradial circle patterns with Dirichlet or Neu-mann boundary conditions in Sections 5.1 and 5.2 respectively. For a certain class of qua-sicrystallic circle patterns, the additional regularity is then used to prove C^∞-convergence in Section 5.3.

5.1 C¹-convergence for Dirichlet boundary conditions

Theorem 5.1. Let D⊂Cbe a simply connected bounded domain, and letW ⊂Cbe open with D ⊂ W. Let g : W → C be a locally injective holomorphic function. Assume, for convenience, that 0∈D.

Forn∈N letDn be a b-quad-graph with corresponding graphs Gn andG^∗_n constructed as above and letαn be an admissible labelling. We assume that Dn is simply connected and that αn is uniformly bounded in the sense that for alln∈Nand all facesf ∈F(Dn)

|αn(f)−π/2|< C (5.1)

with some constant 0< C < π/2 independent ofn.

Let εn ∈(0,∞) be a sequence of positive numbers such that εn → 0 for n→ ∞. For each n ∈ N, assume that there is an isoradial circle pattern for Gn and αn, that is all circles have the same radius εn. Assume further that all centers of circles lie in the domain D and that any point x ∈ D which is not contained in any of the disks bounded by the circles of the pattern has a distance less than Cεˆ n to the nearest center of a circle and to the boundary ∂D, where C >ˆ 0 is some constant independent of n. Denote by Rn ≡ εn

and φn the radius and the angle function of the above circle pattern for Gn and αn. By abuse of notation, we do not distinguish between the realization of the circle pattern, that is the centers of circlesz_n, the intersection pointsv_n and the edges connecting corresponding points in Dn or Gn, and the abstract b-quad-graph Dn and the graphs Gn and G^∗_n. Also, the index nwill be dropped from the notation of the vertices and the edges.

Define another radius function on Gn as follows. At boundary verticesz∈V∂(Gn)set

rn(z) =Rn(z)|g⁰(z)|. (5.2)

Using Theorem 2.10, we can extend rn to a solution of the Dirichlet problem onGn. Let z0∈V(Gn)be such that the disk bounded byCz0 contains0 and lete= [z0, v0]∈E(Dn)be one of the edges incident toz0 such that φn(~e)∈[0,2π)is minimal.

Let ϕn be the angle function corresponding torn that satisfies

ϕn(~e) = arg (g⁰(v0)) +φn(~e). (5.3) Let Cn be the planar circle pattern with radius functionrn and angle functionϕn. Suppose that Cn is normalized by a translation such that

pn(v0) =g(v0), (5.4)

where pn(v)denotes the intersection point corresponding to v∈V(G^∗_n).

77

Forz∈D set

gn(z) =pn(w) and qn(z) = rn(v)

Rn(v)e^{i(ϕn(−vw)−φ→ n(−vw))→} ,

wherewis a vertex ofV(G^∗_n)closest toz andv is a vertex of V(Gn)closest toz such that [v, w]∈E(Dn).

Thenqn→g⁰ andgn→g uniformly on compact subsets in D asn→ ∞.

Remark 5.2.The proof of Theorem 5.1 actually shows the following a priori estimations for the approximating functionsqn andgn.

kqn−g⁰k_V(Gn)∩K ≤C1(−log₂εn)⁻¹² and kgn−gk_V(Gn)∩K ≤C2(−log₂εn)⁻¹² for all compact sets K ⊂ D, where the constants C1, C2 depend on K, g, D and the constants of Theorem 5.1.

We begin with an a priori estimation for the quotients of the radius functions.

Lemma 5.3. Forz∈V(Gn)set

hn(z) = log|g⁰(z)|, tn(z) = log(rn(z)/Rn(z)).

Then

hn(z)−tn(z) =O(εn).

Here and below the notations1 =O(s2) means that there is a constant C which may depend on W, D, g but not on n and z, such that |s1| ≤ Cs2 wherever s1 is defined. A direct consequence of Lemma 5.3 is

rn(z) =Rn(z)|g⁰(z)|+O(ε²_n). (5.5) Our proof uses ideas of Schramm’s proof of the corresponding Lemma 9.2 in [67].

Proof. Consider the function

p(z) =tn(z)−hn(z) +β|z|²,

where β ∈ (0,1) is some function of εn. We want to choose β such that p will have no maximum inVint(Gn).

Suppose that p has a maximum at z ∈ Vint(Gn). Denote by z1, . . . , zm the incident vertices ofz inG_n in counterclockwise order. Then forj= 1, . . . , m we have

tn(zj)−tn(z)≤xj (5.6)

where

xj=hn(zj)−hn(z)−β|zj|²+β|z|². (5.7) First, we gather a little information about thexjs. Sincez∈Vint(Gn), we have|z|=O(1) and by assumptionz−zj =O(εn). Withβ ∈(0,1) this leads toβ|zj|²−β|z|² =O(εn).

Using this estimate and the smoothness of Re(logg⁰), we getxj=O(εn).

From (5.6), the definition oftn(z) = logrn(z)−logRn(z) and the monotonicity of the sum in equation (2.2) (see Lemma 2.7), we get

0 =



X^m

j=1

fα(z,zj)(logrn(zj)−logrn(z))



−π

=



 Xm

j=1

f_α(z,zj)(tn(zj)−tn(z) + log(Rn(zj)/Rn(z)))



−π

≤



 Xm

j=1

f_α(z,zj)³

xj+ log³

Rn(zj) Rn(z)

´´

−π. (5.8)

The preceeding reasonings also apply for general circle patterns with radius functionRn(not necessarily isoradial) which approximate D. In the following, we exploit special properties of the given isoradial circle patterns, that is forRn≡εn.

Rememberingxj=O(εn), we can consider a Taylor expansion of the right hand side of inequality (5.8) about log¡_Rn(zj)

Rn(z)

¢= 0 in order to make anO(ε³_n)-analysis.

Consider the chain of faces fj of Dn (j = 1, . . . , m) which are incident to z and zj. The enumeration of the vertices zj (and hence of the faces fj) and of the black vertices v1, . . . , vmincident to these faces can be chosen such thatfj is incident tovj−1andvj for j = 1, . . . , m, wherev₀=v_m. Furthermore, using this enumeration we have

zj−z

|zj−z|i= vj−vj−1

|vj−vj−1|, (5.9)

see Figure 5.1. Moreover, each face fj of an isoradial circle pattern is a rhombus. So we can write, using the notations of Figure 5.1

zj−z=aj−1+aj and vj−vj−1=aj−aj−1.

Denoting

i i

y

y

©©*

©©©© HHHHjHH HHj

HHHH ©©©©*©© αj

αj

z zj

vj−1

vj

aj

aj−1

aj−1

aj

Figure 5.1: A rhombic face ofDn with oriented edges.

αj =α([z, zj]),

l_j =|z_j−z|= 2R_nsin(α_j/2), ˆlj =|vj−vj−1|= 2Rncos(αj/2),

whereRn≡εnis the constant radius function of the given isoradial circle pattern, we easily obtain by a simple calcu-lation that

fαj(0) = (π−αj)/2, f_α⁰_j(0) = ˆlj/(2lj), f_α⁰⁰_j(0) = 0.

Taking into account that equation (2.2) holds withRn ≡εn

and using the uniform boundedness (5.1) of the labelingα, inequality (5.8) yields 0≤

Xm

j=1

f_α⁰_j(0)xj+O(ε³_n). (5.10) To evaluate this sum, expand

logg⁰(zj)−logg⁰(z) =a(zj−z) +b(zj−z)²+O(ε³_n)

and

xj =hn(zj)−hn(z)−β|zj|²+β|z|²

= Re(a(zj−z) +b(zj−z)²−2β¯z(zj−z))−βl²_j+O(ε³_n).

Noting thatf_α⁰_j(0)(zj−z) = (vj−vj−1)/(2i) we get Xm

j=1

f_α⁰_j(0)xj= Re Ã

a−2βz¯ 2i

Xm

j=1

(vj−vj−1) + b 2i

Xm

j=1

(vj−vj−1)(zj−z)

!

−β Xm

j=1

ljˆlj

2 +O(ε³_n)

= Re Ã

b 2i

Xm

j=1

(aj−aj−1)(aj+aj−1)

!

−β Xm

j=1

ljˆlj

2 +O(ε³_n)

=−β Xm

j=1

ljˆlj

2 +O(ε³_n)

Thus we arrive at 0≤ −βε²_n

Xm

j=1

sin(αj/2) cos(αj/2) +O(ε³_n) ⇐⇒ β Xm

j=1

sin(αj)≤ O(εn).

Note thatε²_nP_m

j=1sin(αj)> πε²_n is the area of the rhombic faces incident to the vertex z.

We conclude from this estimate that

β=O(εn).

This means, that if we chooseβ =Cεn withC >0 a sufficiently large constant and ifεnis small enough such thatCεn<1, thenpwill have no maximum inVint(Gn). In that case, as we havep(z) =β|z|² =O(εn) onV∂(Gn), we deduce that p(z)≤ O(εn) inV(Gn) and thus

tn(z)−hn(z)≤ O(εn) forz∈V(Gn). (5.11) The proof for the reverse inequality is almost the same. The only modifications needed are reversing the sign ofβ and a few inequalities.

Remark 5.4.The statement of Lemma 5.3 can be improved to

hn(z)−tn(z) =O(ε²_n) (5.12) in the case of a ’very regular’ isoradial circle pattern. These are isoradial circle patterns such that for each oriented edge ej1 = zj1 −z ∈ E(G) incident to an interior vertex~ z ∈ V(G) there is another parallel edge ej2 = zj2 −z ∈ E(G) with opposite direction~ incident to z, that is ej2 = −ej1. Furthermore, the corresponding intersection angles agree: α([z, zj1]) =α([z, zj2]). This additional regularity property holds for example for an orthogonal circle pattern with the combinatorics of a part of the square grid.

The proof of estimation (5.12) follows the same reasonings as above, but makes an O(ε⁴_n)-analysis. In particular, we obtain

Xm

j=1

f_α⁰_j(0)xj+ Xm

j=1

f_α⁰⁰⁰_j(0)x³_j+O(ε⁴_n)

=−β Xm

j=1

ljˆlj

2 + Re Ã_m

X

j=1

f_α⁰_j(0)c(zj−z)³

! +

Xm

j=1

f_α⁰⁰⁰_j(0) (Re(a(zj−z)))³+O(ε⁴_n),

where

xj = Re(a(zj−z) +b(zj−z)²+c(zj−z)³−2βz(z¯ j−z))−βl²_j+O(ε⁴_n).

The additional regularity implies that all terms of orderO(ε³_n) vanish.

Remark 5.5.Remember the definition of the discrete Laplacian in (3.2) by

∆η(z) = X

[z,zj]∈E(G)

2f_α([z,z⁰ _j])(0)(η(zj)−η(z)).

The proof of Lemma 5.3 actually shows, that the difference tn−hn is almost harmonic.

More precisely, we have ∆(tn−hn) =O(ε³_n). Adding a suitable subharmonic functionβ|z|² with β >0, we deduce that the resulting functionpis subharmonic, that is ∆p≥0, such thatpattains its maximum at the boundary. This reasoning will also be important for the proof of the following lemma.

Lemma 5.6. Let tn and hn be defined as in Lemma 5.3. Let K ⊂ D be a compact subset in D. Then the following estimation holds for all n∈N and every interior vertex z∈Vint(Gn)∩K such that all its incident verticesz1, . . . , zl are also inVint(Gn)∩K:

tn(zj)−hn(zj)−(tn(z)−hn(z)) =O(εn(−logεn)⁻¹²). (5.13) forj= 1, . . . , l. The constant inO-notation may depend onK, but not on norz.

The proof of Lemma 5.6 uses the following estimation for superharmonic functions, which is a version of Corollary 3.1 of [65]; see also [65, Remark 3.2 and Lemma 2.1].

Proposition 5.7 ([65]). Let G be an undirected connected graph without loops and let c :E(G)→ R⁺ be a positive weight function on the edges. Denote c(e) = c(x, y) for an edge e= [x, y]∈E(G)and assume that

m= max

[x,y]∈E(G)

X

[x,z]∈E(G)

c(x, z) c(x, y) <∞.

Denote by d(x, y)the combinatorial distance between two vertices x, y∈V(G)in the graph G. LetBx(r) ={y ∈V(G) :d(x, y)≤r} be the combinatorial ball of radiusr >0 around the vertexx∈V(G). Fixx∈V(G)andR≥4 and set

A= sup

1≤r≤Rr⁻²Wx(r), where Wx(r) = X

z∈Bx(r), y∈V(G) d(x,z)<d(x,y)

c(z, y).

Let ube a positive superharmonic function in Bx(R+ 1), that is X

[z,w]∈E(G)

c(z, w)(u(z)−u(w))≤0 for all w∈Bx(R+ 1). Lety be incident to xinG. Then

¯¯

¯¯u(x) u(y) −1

¯¯

¯¯≤ 4m²√ p A

c(x, y) log₂R.

Proof of Lemma 5.6. Lemma 5.3 implies thattn(zj)−tn(z) =O(εn) for all incident vertices z, zj∈V(Gn) sincehn= log|g⁰|is aC^∞-function. Consider a Taylor expansion about 0 of

0 = Ã_m

X

j=1

f_α(z,zj)(tn(zj)−tn(z))

!

−π.

Similar reasonings as in the proof of Lemma 5.3 imply that

∆tn(z) = Xm

j=1

2f_α(z,z⁰ _j)(0)(tn(zj)−tn(z)) =O(ε³_n).

Letp=tn−hn+β|z|² withβ ∈(0,1). From the estimations in the proof of Lemma 5.3 we deduce that

∆p(z) = Xm

j=1

2f_α(z,z⁰ _j)(0)(tn(zj)−tn(z))− Xm

j=1

2f_α(z,z⁰ _j)(0)(hn(zj)−hn(z)−β|zj|²+β|z|²)

= 2βε²_n Xm

j=1

sinα([z, zj]) +O(ε³_n).

Thus, if we choose β =Cε_n with C >0 a sufficiently large constant, then ∆p(z)≥0 for all interior verticesz∈Vint(Gn). So fix such a suitableC and define the positive function

ˆ

p=εn+kpk −p.

Then ∆ˆp(z) ≤ 0 for all z ∈ Vint(Gn). The proof of Lemma 5.3 shows that there is a constant C1, depending only ong, D, and the labelling α, such that kpk ≤C1εn. Thus kpk ≤ˆ C2εn withC2= 2C1+ 1.

To finish to proof, we apply Proposition 5.7 to the superharmonic function ˆp. Remember that Gn is a connected graph without loops and c(e) := 2f_α(e)⁰ (0) >0 defines a positive weight function on the edges. The bound (5.1) on the labellingαimplies that

m= max

[x,y]∈E(Gn)

X

[x,z]∈E(Gn)

c(x, z)

c(x, y)< 2π π/2−C

cot(π/4−C/2)

cot(π/4 +C/2) =:C3<∞.

Hence m is uniformly bounded from above for all n ∈ N and the bound C3 depends only on the constant C of (5.1). Let x ∈ Vint(Gn). To find an upper bound for A = sup_1≤r≤Rr⁻²Wx(r), note that

Wx(r) = X

z∈Bx(r),y∈V(Gn) d(x,z)<d(x,y)

c(z, y)≤ max

e∈E(Gn)c(e)#Fw(x, r),

whereFw(x, r) is the set of all faces ofDn with one white vertexz∈Bx(r) and #Fw(x, r) denotes the number of faces ofF_w(x, r). Now, max_e∈E(Gn)c(e)<cot(π/4−C/2)/2 and

Fw(x, r)⊂Dx((r+ 1)2εn) ={w∈C:|w−x| ≤(r+ 1)2εn},

as the edge lengths inG_n are smaller than 2ε_n. Remember thatF(f) =ε²_nsinα(f) is the area of the face f ∈F(Dn). ThusF(f)> ε²_nsin(π/2−C) and

#F_w(x, r)< π((r+ 1)2εn)²

ε²_nsin(π/2−C) ≤ 16πr²

sin(π/2−C)=:r²C₄

for allr≥1. Therefore we obtainA= sup_1≤r≤Rr⁻²Wx(r)< C4, where the upper bound C4 is independent ofR andn∈Nand depends only on the constantC of (5.1).

Let K ⊂ D be compact. Denote by e the Euclidean distance (between a point and a compact set or between closed sets of R² ∼= C) and let ˆC be the same constant as in Theorem 5.1. Asεn→0, we can choosen0∈Nsuch that for alln≥n0we have

Kεn:={z∈C:e(z, K)≤2εn} ⊂Dn⊂D, a(ε_n) :=e(∂D, K)−(2 + ˆC)ε_n >0,

C5:= min

n≥n0

a(εn)

2 cos(π/2−C/2) ≥ε²_n, and εn≤C5/4.

Let z ∈V(Gn)∩K and set R+ 1 = d(z, V∂(Gn)) to be the combinatorial distance from z to the boundary of G_n. Let z_j ∈ V(G_n) be incident to z. Then z_j ∈ K_εn and R ≥ d(zj, V∂(Gn)). As the Euclidean distance e(z1, z2) = 2εncos(α([z1, z2])/2) between two vertices ofGn is smaller than 2εncos(π/4−C/2), we obtain

R≥ e(zj, V∂(Gn)) 2εncos(π/4−C/2). Nowzj ∈Kεn implies

e(z_j, V_∂(G_n))≥e(K_εn, V_∂(G_n))≥e(K_εn, ∂D)−Cεˆ _n≥e(∂D, K)−(2 + ˆC)ε_n=a(ε_n)>0 forn≥n0. ThusR≥ε⁻¹_n a(εn)/(2 cos(π/2−C/2))≥ε⁻¹_n C5≥4 and

p 1

log₂R ≤ 1

plog₂C5−log₂εn

≤

√2 p−log₂εn

hold for alln≥n0 and allz∈V(Gn)∩K by our assumptions. Proposition 5.7 implies

¯¯

¯¯p(z)ˆ ˆ

p(zj)−1

¯¯

¯¯≤ 4C₃²√ C₄√ p 2

−c(z, zj) log₂εn

for all incident verticesz, zj∈V(Gn)∩Kandn≥n0. Asc(z, zj)≤cot(π/4−C/2)/2 and kpk ≤ˆ C2εn we finally arrive at the desired estimation

|tn(zj)−hn(zj)−(tn(z)−hn(z))|=|ˆp(z)−p(zˆ j)| ≤C6εn(−log₂εn)⁻¹²

for all incident verticesz, zj∈V(Gn)∩K andn≥n0, where the constantC6 depends on the previous constants C2, . . . , C5, that isC6depends only on g,D,C, ˆC, andK.

Lemma 5.8. Let ~e =−uv→ ∈ E(D~ n) be a directed edge with u∈ V(Gn) and v ∈ V(G^∗_n).

Denote byδn(e)the combinatorial distance in graphDn frome= [u, v]to[z0, v0], that is the least integerksuch that there is a sequence of edges{[z0, v0] =e1, e2, . . . , ek=e} ⊂E(Dn) such that the edges em+1 andem are incident to the same face inDn form= 1, . . . , k−1.

Then

ϕn(~e) = argg⁰(v) +φn(~e) +δn(e)O(εn(−log₂εn)⁻¹²). (5.14) The constant in the notation O(εn(−log₂εn)⁻¹²) may depend on the distance of v to the boundary ∂D.

Note that if∂D is smooth, then δn(e) =O(ε⁻¹_n ). In general we have δn(e) =O(ε⁻¹_n ) on compact subsetsK⊂D, where the constant in the notationO(ε⁻¹_n ) may depend onK.

In any case, on compact subsets ofD we have

ϕn(~e) = argg⁰(v) +φn(~e) +O((−log₂εn)⁻¹²). (5.15) Proof. We use estimation (5.13) and Theorem 2.17.

Using the notation of Figure 2.6 (left), equation (5.13) implies fα(logrn(z+)−logrn(z−)) =fα(log rn(z+)

Rn(z+)−log rn(z−) Rn(z−))

=fα(log|g⁰(z+)| −log|g⁰(z−)|) +O(εn(−log₂εn)⁻¹²)

=fα(0) +f_α⁰(0)(log|g⁰(z+)| −log|g⁰(z−)|) +O(εn(−log₂εn)⁻¹²).

Im Dokument Approximation of conformal mappings by circle patterns and discrete minimal surfaces (Seite 75-91)